MAGAZINE PRICES REVISITED
Jonathan L. Willis
DECEMBER 2001
RWP 01-15
Research Division
Federal Reserve Bank of Kansas City
Jonathan L. Willis is an economist at the Federal Reserve Bank of Kansas City. This paper is a
revised version of the first chapter of the author’s Ph.D. dissertation. The author would like to thank
Russell Cooper for detailed discussions regarding problems of unobserved heterogeneity and state dependence and Stephen Cecchetti for providing the magazine data. Kent Kovacs provided excellent research
assistance. The views expressed in this paper are solely those of the author and do not necessarily reflect
the views of the Federal Reserve Bank of Kansas City or the Federal Reserve System.
Willis email: [email protected]
Abstract
This paper examines price adjustment behavior in the magazine industry. In a frequently cited
study, Cecchetti (1986) constructs a reduced-form (S,s) model for firms. Cecchetti assumes that a firm's
pricing rules are fixed for non-overlapping three-year intervals and estimates the model using a conditional
logit specification from Chamberlain (1980). The estimates are inconsistent, however, due to the statedependent specification of the model. I illustrate the econometric problems in Cecchetti's results through a
Monte Carlo exercise and then suggest a method for producing consistent estimates based upon Heckman
and Singer (1984). The corrected results provide strong support for models of state-dependent pricing.
JEL classification: C14, D40, and L16
Keywords: menu costs, price adjustment, nonparametric methods
1
Introduction
In a frequently cited study of price adjustment, Cecchetti (1986) uses data on magazine
cover prices to examine determinants of the frequency of price adjustment. Based upon
his empirical results, Cecchetti makes two conclusions: 1) higher inßation leads to more
frequent price adjustment and 2) the real cost of a nominal price change varies with either
the frequency of adjustment or the size of a real price change. The empirical techniques used
by Cecchetti, however, are inconsistent due to the presence of state-dependent covariates
in the estimation. This paper will outline the econometric problems inherent in Cecchetti’s
results, test for various forms of heterogeneity in the magazine data, and offer a procedure
to obtain consistent estimates. The corrected results conÞrm Cecchetti’s Þrst conclusion —
providing strong evidence for models of state-dependent pricing — but do not yield evidence
on the structure of price adjustment costs.
Cecchetti’s paper is one of the few empirical studies of price determination that attempts
to estimate a model using microeconomic data.1 A primary reason for the lack of empirical
work is the difficulty in obtaining transaction price data. The data on magazine cover
prices, in addition to being easily collectable, possess the desired characteristics for the
study of price changes. Magazine cover prices change at discrete intervals, remain Þxed for
long periods of time, and are not the result of an auction process. However, one potential
drawback is that the magazine industry does not rely solely on revenue generated through
transactions occurring at the cover price. Subscription sales and advertising fees provide a
large portion of the revenues for the industry.2
Cecchetti addresses two questions stemming from the theoretical literature of price de1
Recent papers focusing on the estimation of price adjustment costs include Aguirregabiria (1999), Slade
(1998), and Willis (2000).
2
Subscription sales account for about 70 percent of circulation on average over the sample period.
In recent years advertising revenue has accounted for over 70 percent of total revenue for magazines in
Cecchetti’s sample.
2
termination for monopolistically competitive Þrms. The Þrst concerns the response of the
frequency of adjustment to increases in general price inßation. Theory demonstrates that
Þrms will adjust by larger amounts when faced with higher inßation, holding Þxed the size
of the adjustment costs, but theoretical models do not point to a clear relationship between
inßation and the frequency of adjustment.3 The second question pertains to the appropriate structure of the cost of price adjustment. Several theoretical studies have examined the
size of a price adjustment cost necessary for a nominal demand shock to produce a certain
response for real variables.4 Providing empirical evidence on price adjustment costs, however, is much more challenging due to the lack of industry data and precise cost models.
Obtaining estimates of these adjustment costs would provide useful evidence on the degree
of price rigidity for Þrms. In addition, these estimates of adjustment costs could be used
to study the aggregate implications of monetary shocks for real output.
The foundation of Cecchetti’s empirical analysis is based on a target-threshold model
from Iwai (1981). This model employs threshold barriers to signal when the price should
be changed. If the optimal price surpasses a Þxed distance from the current setting, then
the price is adjusted. Prices are not constantly updated because of the presence of “menu
costs,” which could relate speciÞcally to the physical costs of implementing price changes
or to a broader variety of costs associated with information gathering costs and managerial
costs necessary for the Þrm to make an fully informed optimization decision.
The Þrm’s probability of a price change is derived from the optimal pricing decision.
The probability can be expressed as a function of the change in the optimal price and the
distance between the previous return point and the current threshold barrier of the (S, s)
model. An analytic solution for the change in the optimal price, expressed in terms of
observable variables, is obtained, but the distance between optimal (S, s) barriers is not
observed. This unobserved “variable” represents the optimal pricing rules set by Þrms
3
4
See Sheshinski and Weiss (1977,1983).
See Blanchard and Kiyotaki (1987) and Ball and Romer (1990).
3
each period, which are inherently functions of the state variables of each Þrm’s dynamic
problem. Cecchetti identiÞes the model by assuming that the optimal pricing rules are
constant over non-overlapping three-year increments. This assumption leads to a Þxedeffects speciÞcation to control for the unobserved heterogeneity resulting from Þrm-speciÞc
optimal pricing rules. Due to the small number of observations per Þxed-effect group (3
annual observations), Cecchetti uses a conditional logit estimation procedure described by
Chamberlain (1980) to account for the presence of heterogeneity. However, in a subsequent
paper, Card and Sullivan (1988) show that the presence of state-dependent covariates,
which are included in Cecchetti’s speciÞcation, leads to inconsistent estimates.5 The loss of
consistency nulliÞes the results of Hausman tests used to support Cecchetti’s identiÞcation
assumption.
I obtain consistent estimates using the approach of Heckman and Singer (1984). This
procedure is based upon a random-effects speciÞcation, where the distribution of individual
effects is assumed to be discrete. Within the estimation procedure, the discrete mass point
distribution is estimated along with the other coefficients of interest. I perform two Monte
Carlo exercises to illustrate the beneÞts of this approach. When applied to the magazine
data, however, this procedure does not produce results signiÞcantly different from the
unconditional logit model with no Þxed effects
A Þnal model is estimated based upon an alternative speciÞcation of heterogeneity. I
specify a Þxed-effects logit model with dummy variables to control for aggregate year effects
and heterogeneity across magazines, rather than across magazine-speciÞc 3-year intervals.
These estimates provide strong support for models of state-dependent pricing, but they do
not address the question concerning the structure of price adjustment costs.
5
While a subset of econometricians are well-aware of this result, in general practice the problems intro-
duced by state dependence are not often recognized. For example, Greene (1999) cites Cecchetti (1986) as
a good empirical application of Chamberlain’s conditional logit estimation procedure.
4
2
Model
In specifying a model for the pricing decision of a monopolistically competitive Þrm, the
presence of price-adjustment costs and future uncertainty indicate the need for a dynamicprogramming framework. In such a model, a Þrm chooses its price by maximizing the
expected present discounted value of current and future proÞts, factoring in the price
adjustment cost to be paid if adjustment occurs. Sheshinski and Weiss (1983) specify
a general set of conditions where it is equivalently optimal for the Þrm to use a targetthreshold (S, s) policy where adjustment occurs when the Þrm’s price crosses an optimal
barrier.
Cecchetti uses a target-threshold model described by Iwai (1981). In this model, Þrms
develop a rule dictating that the price at time t, Pt , should be changed when it is a certain
distance from the short-term optimal price, Pt∗ . The short-term optimal price is deÞned
as the price that would be set without adjustment costs. DeÞne zt = log(Pt∗ /Pt ) as the
measure of the distance between the actual price and optimal price. Let hct be the maximum
value zt can reach before the price will be changed and h0t be the prescribed value of zt
to be attained when the price change occurs. The optimal reset value, h0t , may be equal
to zero, but if prices tend to drift upward, the Þrm will likely choose to raise the price
above the optimal price in order to reduce the number of adjustments, thereby reducing
the incidence of paying adjustment costs. In this situation, h0t would be negative.
The probability of observing a price change can be written as the probability of zt rising
above hct , indicating that the distance between the current price and the optimal price has
surpassed the limit established by the (S, s) rule. This probability can also be rewritten
in terms of comparing the distance Pt∗ has moved since the last price change at time t̃
to the distances between the previous return point, h0t̃ , and the current cutoff rule, hct .6
6
Recall that zt = log Pt∗ − log Pt . Pt was set in period t̃ according to the reset value h0t̃ = log Pt̃∗ − log Pt .
Therefore, we can substitute for zt using log Pt∗ − log Pt̃∗ + h0t̃ .
5
The problem with this setup, noted by Cecchetti, is that the rule should change over time
as the Þrm reoptimizes. Assumptions about the timing of rule changes will lead to the
speciÞcation of unobserved heterogeneity in the estimation.
For Þrm i, the probability of a price change at time t is as follows:
¡
¢
Pr(yi,t = 1) = Pr zi,t ≥ hci,t
´
³
∗
c
0
≥
h
−
h
= Pr ∆ log Pi,t,
i,t
t̃
i,t̃
(1)
∗
where yi,t is a dummy variable equal to 1 if Þrm i changed its price at time t, ∆ log Pi,t,
is
t̃
the cumulative change in the optimal price since the last price change at time t̃, and h0i,t̃ is
the return point used at the time of the previous price change.
To solve for P ∗ , each Þrm is assumed to be a monopolistic competitor with the following
¡
¢
demand Qdi,t and cost functions (C (Qi,t )):
Qdi,t
=
µ
Pi,t
P̄t
¶b
Xtγ
C (Qi,t ) = Aeδt Qαi,t wt
(2)
(3)
where Qi,t represents Þrm output, Pi,t is the Þrm’s price, P̄t is the aggregate price level, Xt
is total industry sales, δ is rate of technological progress, wt is the input price, and b, γ, A,
and α are constants. The demand and cost equations are substituted into the Þrm’s proÞt
function. Maximization of this function with respect to the price allows the calculation of
∗
log Pi,t
. Cecchetti assumes that Pt and wt increase at the same constant rate and derives
the change in optimal prices as
∗
ú
∆ log Pi,t,
t̃ = b1 Ti,t,t̃ + b2 πi,t,t̃ + b3 Xi,t,t̃ + ui,t
(4)
where Ti,t,t̃ is the time since the previous price change for magazine i, πi,t,t̃ is the cumulative
inßation since the last price change, Xú i,t,t̃ is the cumulative percentage change in industry
sales since the last price change, and ui,t is a stochastic error. Cecchetti states that the
6
stochastic error term represents components of the optimal pricing decision that are not
contained in the demand and cost functions.
Substitution into equation (1) yields
³
´
∗
c
0
Si,t = ∆ log Pi,t,
−
h
−
h
i,t
t̃
i,t̃
³
´
c
0
ú
= b1 Ti,t,t̃ + b2 πi,t,t̃ + b3 Xi,t,t̃ + ui,t − hi,t − hi,t̃
(5)
By assuming that ui,t has a logistic distribution, the estimating equation is
¡ ¢
Pr (yi,t = 1) = F S̄i,t
(6)
where F is the logistic function and S̄i,t = Si,t − ui,t .
3
Methods
³
´
In order to estimate equation (6), the unobserved optimal pricing rules hci,t − h0i,t̃ must
be identiÞed. These rules represent the policy functions that solve the Þrm’s dynamic
optimization problem, and as such, they are implicit functions of the state variables in the
Þrm’s current information set. As the information set is updated over time, these rules will
necessarily by modiÞed as the Þrm reoptimizes. If these pricing rules are not accounted for
in the estimation, the results will be subject to standard omitted variable bias.
Cecchetti argues that in the short run, a Þrm’s optimal pricing rules are largely based on
long-term expectations formed over several previous years. Based upon this interpretation,
his identifying assumption is that pricing rules for a magazine are Þxed in non-overlapping
three-year increments. He can therefore treat the pricing rules as a Þxed effect in his
estimation procedure. In some cases, a Þxed effect can be controlled for in estimation
through differencing or dummy variables. Unfortunately, differencing does not lend itself
to nonlinear models of the type here. Chamberlain (1980) illustrates a second problem,
related to the size of the sample. In a panel data set where the number of individuals (N )
7
is large and number of observations per individual (T̄ ) is small, nonlinear models do not
possess the same consistency property (for Þxed T̄ ) as the linear Þxed-effect model. For
a dataset in which there are 2 observations per individual, Chamberlain shows that in a
simple model with one covariate, the coefficient estimate using a logit distribution will be
double the true value.
3.1
Cecchetti’s estimation procedure
Cecchetti estimates his model using a method proposed by Chamberlain (1980). Chamberlain describes a Þxed-effects speciÞcation for situations in which the number of observations
per individual is small. Using the logistic function, the sum of the dependent variable over
the sample for an individual serves as a sufficient statistic for the Þxed-effects term. By
conditioning on this sum, the Þxed-effect term falls out of the conditional probability. The
conditional likelihood function is then constructed as the product of the individual probabilities and maximized in nonlinear fashion. The estimates are consistent for data in
which T̄ is greater than or equal to two. The disadvantage of this estimation technique
is that the Þxed-effects coefficients are not estimated. Also, whenever the conditioning
summation equals zero or T̄ , the probability for the individual is degenerate. Thus, the
results of the conditional Þxed-effects speciÞcation cannot be directly compared to results
of unconditional logit estimates.
The inconsistency of Cecchetti’s estimates arises due to the presence of state-dependent
covariates. The cumulative variables measuring the number of years since the last price
¢
¡
change Ti,t,t̃ , more commonly referred to as duration in the literature; cumulative inßation
¡
¢
since the last price change πi,t,t̃ ; and cumulative percentage change in sales since the
³
´
ú
previous change Xi,t,t̃ , are all functions of lagged dependent variables as their values
are based on the residing states in the past.7 Card and Sullivan (1988) have shown that
7
For example, the duration term can be expressed as Ti,t,t̃ = (1 − yi,t−1 ) ∗ Ti,t−1,t̃ + 1.
8
the conditional likelihood approach cannot be extended to the logistic model when one of
the explanatory variables is state dependent. The minimal sufficient statistic for the Þxed
effect, in all but a few exceptional cases, is the entire set of data.8 Thus, the Chamberlain
speciÞcation, conditioning on the sum of the dependent variable for the individual, will not
provide consistent estimates for the determinants of price adjustment.
The lack of consistency invalidates Hausman tests used by Cecchetti to support his
model speciÞcation over several alternatives. The two alternative models tested contain
simpler heterogeneity assumptions than described in the model above. One model contains a single constant term, assuming that there are no Þxed effects across Þrms or over
time. The second model includes a dummy variable for each magazine Þrm. Both of
these alternative models are nested within Cecchetti’s model. The difference is that the
two alternative models can be estimated using the unconditional likelihood function, while
Cecchetti estimated his model using Chamberlain’s conditional likelihood method due to
the small number of observations per “individual.” The Hausman procedure tests whether
the parameter vectors of two nested models are signiÞcantly different under the null that
both models are consistent and the model with fewer parameters is efficient. Since the
Chamberlain Þxed-effect speciÞcation is not consistent, this test cannot be used to reject
a simpler structure.
3.2
Heckman-Singer method
To obtain consistent estimates, I directly account for the individual-speciÞc “variable”
in the unconditional likelihood function using a random-effects speciÞcation to model Cecchetti’s assumption that policy rules do not change over non-overlapping three year periods.
Several methods exist where the distributional shape of an individual effect is assumed, and
8
The exceptional cases consist of situations where the state-dependent covariates are not functions of
the relevant state variables for the policy function, i.e. the coefficients for these variables in the logistic
function are equal to zero.
9
then the effect is integrated out of the likelihood function. Heckman and Singer (1984) propose a procedure that abstracts from the assumption of a speciÞc parametric representation
of the distribution of the random effect by allowing for a partial parametric speciÞcation.
This speciÞcation allows the unknown distribution to be represented non-parameterically
by a step function. In this manner the probability density function is approximated by
a discrete distribution with a certain number of mass points, and estimates are made for
the location and density of each point. Lindsay (1983a, 1983b) shows that for this type of
problem, the maximum of the likelihood function will contain a distribution with a Þnite
number of mass points. The precise number of mass points can be determined by beginning
with one support (i.e. no heterogeneity) and working upward. The location of an additional
mass point is determined by Þnding the value that maximizes the Gauteaux derivative.9 If
the maximal value of the derivative is less than or equal to zero, then no more mass points
are necessary. Although the distribution of the individual effects is not likely to be well
characterized by the step function, Heckman and Singer have shown through Monte Carlo
experiments that the coefficients of the other explanatory variables can be estimated with
great precision.
3.3
Monte Carlo exercise
A Monte Carlo study illustrates the problems induced by state dependence in Cecchetti’s
estimates and the beneÞts of the Heckman-Singer estimation procedure. The “data” is constructed using Cecchetti’s logit speciÞcation and data on inßation and magazine industry
sales. From equations (5) and (6), the probability of a price change is expressed as
³
´
exp ai,t,t̃ + b1 Ti,t,t̃ + b2 πi,t,t̃ + b3 Xú i,t,t̃
³
´
Pr (yi,t = 1) =
ú
1 + exp ai,t,t̃ + b1 Ti,t,t̃ + b2 πi,t,t̃ + b3 Xi,t,t̃
9
See Heckman and Singer (1984) for details.
10
(7)
³
´
where ai,t,t̃ ≡ − hci,t − h0i,t̃ . The key assumption concerns the speciÞcation of the agent’s
optimal pricing rules, ai,t,t̃ . For this exercise, Cecchetti’s assumption of three-year Þxed
pricing rules is used. The value of ai,t,t̃ for every non-overlapping three-year period is randomly drawn from a uniform distribution in the range [−8, −2], which roughly corresponds
to average values backed-out from Cecchetti’s estimation using magazine data.10 Cecchetti’s parameter estimates are taken as “truth.”11 For this experiment, 10,000 datasets
are generated, each composed of 38 Þrms simulated over 27 years.12 Firm-speciÞc randomly
generated individual effects are selected for each non-overlapping three-year period. The
covariates are created sequentially as a function of the lagged dependent variable and actual data on aggregate inßation and industry sales. I estimate the coefficients using three
different estimation methods: Chamberlain’s Þxed-effects conditional logit, unconditional
logit, and the Heckman-Singer random-effects logit speciÞcation. The results are presented
in Table 1.
Column 1 lists the values used in constructing the simulated data, where “truth” represents parameter estimates from the actual magazine data. The problems caused by
state-dependent covariates in the Chamberlain speciÞcation in column 2 are immediately
evident. Focusing on the coefficient of the duration variable, Ti,t,t̃ , we see that the estimate
is severely skewed upwards; the true value is more than one standard deviation away. In column 3, the unconditional logit estimate for the coefficient on duration, on the other hand,
10
It is important to note that due to the random draws, there is no serial correlation built into the
Þxed effect terms. One might think that the optimal pricing rules should be serially correlated over time
— particularly since this is the basis for Cecchetti’s assumption of Þxed effects. Despite the randomness,
the Þxed effects will be correlated with the covariates due to the state dependence described earlier. If no
correlation existed, then there would be no omitted variable bias. In the appendix, alternative models are
created to examine the impact of serial correlation in the Þxed effects.
11
The parameters used as “truth” come from the replication of Cecchetti’s estimation that will be
discussed below.
12
The size of the sample is chosen to correspond to the size of the magazine panel dataset.
11
is downward biased. This result matches the standard prediction that hazard slopes are
downward biased when heterogeneity is not taken into account. Columns 4 and 5 present
the Heckman-Singer estimates when the distribution representing the individual-speciÞc
effects contains two and three mass points respectively. The duration coefficient estimate
for the speciÞcation with three mass points is very close to the true value. Estimates for
the other coefficients, cumulative inßation and change in sales since the previous change,
are more difficult to interpret because these variables are combinations of the lagged dependent variable and exogenous variables, whereas the duration variable is solely a function
of the lagged dependent variable. The Heckman-Singer estimates in column 5 are reasonably close to the true values and the standard deviations of these simulation estimates are
much smaller than for the conditional logit estimates. For the vast majority of the 10,000
samples, the addition of a fourth mass point did not lead to signiÞcant improvements in
the likelihood function.
A second Monte Carlo experiment focuses on the interaction between the state-dependent
covariates. Each covariate represents a cost or demand inßuence on the pricing decision,
but these terms may often be collinear. For example, technology growth in this model is
represented by the duration variable (Tit ), indicating that technology is assumed to increase
at a constant rate. Inßation is represented by cumulated inßation since last change (πTit ).
In periods with relatively constant inßation, the model will not be able to distinguish between the two inßuences. This point is illustrated by constructing a dataset where the
duration coefficient is set at 0. As before, 10,000 datasets were simulated, but this time
“truth” was altered by setting the coefficient on Ti,t,t̃ equal to 0. The results are shown in
Table 2.
By comparing “truth” in column 1 of Table 2 with the conditional logit estimates in
column 2, it is readily apparent that the Chamberlain estimates suffer severely from state
dependence. The problems from the Þrst exercise are compounded by the inclusion of the
¡
¢
duration variable Ti,t,t̃ when it has no impact on the model. Instead of obtaining an
12
estimate of the duration coefficient near 0, the average estimate is 2.31 and the standard
deviation of the 10,000 simulation estimates is 5.07. The unconditional logit estimates in
column 3 suffer from the standard downward bias associated with ignoring the presence
of heterogeneity. The Heckman-Singer point estimate for the duration coefficient matches
truth precisely, and the coefficient estimates for πi,t,t̃ and Yú i,t,t̃ in column 5 appear to be
slightly upward biased but well within the range of one standard deviation of the true
parameters. We now examine these methods when applied to the magazine panel dataset.
4
Data
The magazine data represent the newsstand prices of thirty-eight magazines over the period
1953 to 1979.13 The frequency of the data is annual, and a price change in a given year is
deÞned as a change in the cover price between the Þrst issue of the given year and the Þrst
issue of the following year. Inßation data is based upon the deßator for gross domestic nonfarm produce, excluding housing services, from the Department of Commerce. Aggregate
single copy sales data for the magazine industry were acquired by Cecchetti from the
Magazine Publishers Association.
5
Results
In addition to replicating the estimates of Cecchetti and providing consistent estimates
using the Heckman-Singer procedure, I perform two other estimations using alternative
identiÞcation assumptions for the unobserved pricing rules. Each speciÞcation is based on
the model presented by the combination of equations (5) and (6). First, the unconditional
logit model is estimated, where the pricing rules are assumed to be the same for all Þrms
and all periods in the sample. Next, a logit model with Þxed-effects dummy variables is
13
The list of magazines is located in the appendix of Cecchetti (1986).
13
estimated where the “individual” is deÞned as a magazine, rather than a three-year span
within a magazine’s history. Then, Chamberlain’s Þxed effects conditional logit procedure is
used to replicate Cecchetti’s results.14 Finally, the Heckman-Singer estimates are presented.
Results are reported in Table 3. The Þxed-effects speciÞcation is denoted as (i = 3-year)
for non-overlapping three-year groups and (i = mag.) for Þxed effects at the magazine level.
The logit estimation results without Þxed effects are in column 1. First, notice that the
duration coefficient is negative, indicating that the probability of adjustment decreases as
the time since the previous change increases. However, as mentioned before, a downward
bias in the duration coefficient is expected in cases where heterogeneity has been ignored.
This result is due to the fact that “individuals” who are very likely to change prices have
done so in the Þrst few years. The remaining “individuals,” as time progresses, are those
who are less likely to change prices. Failing to control for individual differences will cause
the duration parameter to be biased downward because some individuals will be frequently
changing prices after a year or two, over-representing the probability of a change in shorter
durations compared to those who seldom change prices.
In column 2, Þxed-effect dummy variables for individual magazines are added to the
unconditional logit speciÞcation of column 1.15 The coefficient estimate for duration is
greater than the estimate in column 1, although it remains signiÞcantly negative. The
coefficient for cumulative inßation since last change increases, while the coefficient on percentage change in sales falls. In comparing the Þrst two models, the likelihood ratio statistic
14
Cecchetti’s results could not be precisely replicated due to the lack of price data for some magazines.
His results are based upon a subsample of 954 observations. Since I do not have all of the data necessary
to duplicate his subsample, which requires information on earlier price changes, the replication results are
based upon the entire sample of 38 magazines for 27 years. Using Cecchetti’s subsample criteria with the
limited data omits 219 of the 1026 total observations and produces similar results.
15
While using Þxed-effect dummy variables to control for Cecchetti’s assumed form of heterogeneity
where T̄ = 3 would lead to severe bias in the estimates, the bias should be small for a model with magazine
Þxed effects where T̄ = 27.
14
indicates rejection of the restricted model in column 1 at the 5 percent level of signiÞcance
in favor of the Þxed-effects speciÞcation.16 If heterogeneity in the data is best captured by
differences across magazines, then duration appears to have a slight negative relationship
with the probability of a price change.
The results from the conditional logit estimation are contained in column 3. The coefficient estimate for duration is 1.02, much greater than estimates from the other models
and indicative of a time-dependent relationship with the probability of price adjustment.
The estimates of the other coefficients are also much larger than those in columns 1 and
2. The inconsistency of the estimates due to state dependence precludes any signiÞcance
testing.
The Heckman-Singer estimates in column 4 are similar to the estimates in the Þrst two
columns. As discussed above, the Heckman-Singer procedure calls for the gradual addition
of supports (mass points) to the distribution, which are then estimated along with the
probabilities. The reported results are based upon a model in which two mass points are
used to represent the distribution of Cecchetti’s assumed form of individual-speciÞc effects.
The addition of the second mass point resulted in only a marginal improvement of the
likelihood function over the model in column 1, which is the model with a single mass
point. Adding a third mass point resulted in no improvement of the likelihood function.
This contrasts with the Monte Carlo estimation above where three distinct mass points were
estimated and the other coefficient estimates were markedly different from the restricted
logit model estimates.
Based upon the estimates from the Chamberlain conditional logit model, Cecchetti
backs out a time series of average effects in the magazine industry. From this evidence,
he argues that the movements in the effects are inconsistent with a constant real cost of
price adjustment. A similar calculation cannot be computed from the Heckman-Singer
16
The likelihood ratio statistic is 54.5 with 37 degrees of freedom. The 5 percent critical value for the
chi-square distribution is 52.
15
estimates. The limitation of this model is that the effects are not estimated precisely. Even
more problematic is the lack of signiÞcant improvement in the likelihood function over the
logit model, which provides little support for Cecchetti’s identiÞcation assumption in the
Þrst place.
There are several possible explanations for the Heckman-Singer estimation results. One
concerns the issue of how the individual-speciÞc effect terms, ai,t,t̃ , are generated. In the
speciÞcation of the Heckman-Singer unconditional likelihood, I assume that the terms are
randomly generated every 3 years. Despite the randomness, the variable will be correlated
with the covariates because of the state dependence, but it may be more plausible to
think that ai,t,t̃ should be correlated across the three-year intervals for a given magazine.
Incorporating the serial correlation into the Heckman-Singer speciÞcation, however, would
lead to a very complicated likelihood function. Appendix A examines the Heckman-Singer
model estimates under alternative data generating models.
An alternative approach is to assume that the optimal pricing rules are primarily functions of aggregate variables, such as inßation and money growth. This would imply that
there would be Þxed effects across magazines due to aggregate ßuctuations, and these could
be controlled through the introduction of time dummy variables. An additional explanation
is that signiÞcant differences exist across magazines due to unobservable factors, such as
managerial ability. Table 3 provides some evidence for the latter interpretation by comparing the likelihood values of the Þxed effects logit model in column 2 to the unconditional
logit model in column 1.
Combining these two hypotheses, Table 4 reports estimates of a Þxed effects logit model
which includes annual and magazine dummy variables. For comparison purposes, estimates
from the restricted logit model in Table 3 are reproduced in column 1. Based upon the
likelihood-ratio statistic, the restricted model in column 1 is rejected in favor of the model
with Þxed effects in column 2.17 In comparison to the restricted model, two important
17
The likelihood ratio statistic is 157.68 with 63 degrees of freedom. The 1 percent critical value for the
16
results emerge. First, the duration coefficient in now insigniÞcantly different from zero,
indicating that models based upon time-dependent price changes are not likely to closely
Þt the data. Second, the coefficient on cumulative inßation remains positive and statistically
signiÞcant, providing support for state-dependent models.
6
Conclusion
Cecchetti’s analysis is well motivated in its pursuit for empirical results documenting price
stickiness in a particular industry. His selection of the magazine industry as the focus of
analysis is arguably a good choice because of the Þxed nature of magazine cover prices. The
inconsistency of his estimates, however, nulliÞes the tests used to support his speciÞcation
of heterogeneity. The Monte Carlo exercises illustrate the properties of estimates provided
by the Heckman-Singer procedure, but when applied to the magazine data, this procedure
does not identify signiÞcant heterogeneity at the three-year level. Other speciÞcations provide some insight into the structure of heterogeneity. In particular, the estimates indicate
support for heterogeneity at the magazine level. The use of annual dummy variables to
capture the effects of aggregate variables also appears to be beneÞcial.
The corrected results conÞrm Cecchetti’s Þnding that cumulative inßation is a primary
factor in determining the probability of a price change. His conclusions on the behavior
of the real cost of price adjustment and the effect of higher inßation on the frequency
of adjustment, however, are dependent on his inconsistent estimation results. While the
corrected estimates provide some information on the heterogeneity of magazines, this approach is not able to adequately control for the underlying pricing rules in a manner that
permits inference on the structure of price adjustment costs.
An alternative approach would be to directly specify a structural dynamic-programming
relevant chi-square distribution is 92.01.
17
model for a Þrm in this industry.18 A structural model will permit direct analysis of
variously speciÞed sources of heterogeneity and idiosyncratic differences across Þrms. More
importantly, this type of analysis allows for direct estimates of the magnitude of price
adjustment costs.
18
See Willis (2000).
18
References
Aguirregabiria, Victor, “The Dynamics of Markups and Inventories in Retailing Firms,”
Review of Economics Studies, April 1999, 66 (2), 275—308.
Ball, Laurence and David Romer, “Real Rigidities and the Non-Neutrality of Money,”
The Review of Economic Studies, April 1990, 57 (2), 183—204.
Blanchard, Olivier and Nobuhiro Kiyotaki, “Monopolistic Competition and the Effects of Aggregate Demand,” American Economic Review, September 1987, 77, 647—
666.
Card, Daniel and Daniel Sullivan, “Measuring the Effect of Subsidized Training Programs on Movements In and Out of Employment,” Econometrica, May 1988, 56,
497—530.
Cecchetti, Stephen, “The Frequency of Price Adjustment: A Study of the Newsstand
Prices of Magazines,” Journal of Econometrics, August 1986, 31, 255—274.
Chamberlain, Gary, “Analysis of Covariance with Qualitative Data,” Review of Economic Studies, 1980, 47, 225—238.
, “Heterogeneity, Omitted Variable Bias, and Duration Dependence,” Longitudinal
Analysis of Labor Market Data, James Heckman and Burton Singer, eds. Cambridge
University Press: Cambridge, UK 1985.
Greene, William, Econometric Analysis, Macmillan: New York 1999.
Heckman, James and Burton Singer, “The IdentiÞcation Problem in Econometric
Models for Duration Data,” Advances in Econometrics, Cambridge University Press:
Cambridge, UK 1982.
19
and
, “A Method for Minimizing the Impact of Distributional Assumptions in
Econometric Models for Duration Data,” Econometrica, March 1984, 52 (2), 271—320.
Iwai, Katsuhito, Disequilibrium Dynamics, Yale University Press: New Haven, CT 1981.
Lancaster, Tony, The Econometric Analysis of Transition Data, Cambridge University
Press: Cambridge, UK 1990.
Lindsay, Bruce, “The Geometry of Mixture Likelihoods, Part I,” Annals of Statistics,
1983a, 11, 86—94.
, “The Geometry of Mixture Likelihoods, Part II,” Annals of Statistics, 1983b, 11,
783—792.
Sheshinski, Eytan and Yoram Weiss, “Inßation and Costs of Price Adjustment,”
Review of Economics Studies, 1977, 44, 281—303.
and
, “Optimal Pricing Policy Under Stochastic Inßation,” Review of Economics
Studies, 1983, 51, 281—303.
Slade, Margaret, “Optimal Pricing with Costly Adjustment: Evidence from RetailGrocery Prices,” Review of Economic Studies, January 1998, 65, 87—107.
Trussell, James and Toni Richards, “Correcting for Unmeasured Heterogeneity in
Hazard Models: An Application of the Heckman-Singer Strategy to Demographic
Data,” Sociological Methodology, 1985, pp. 242—276.
Willis, Jonathan, “Estimation of Adjustment Costs in a Model of State-Dependent Pricing,” Federal Reserve Bank of Kansas City Research Paper No. 00-07, December 2000.
20
Appendix A
In the Monte Carlo exercises presented in the paper, the Heckman-Singer model produced estimates that were very close to the parameters of the data generating model.
The data generating model, however, contained an assumption on the stochastic nature of
the individual-speciÞc effect that may not be true for the magazine industry. This section
examines the robustness of the Heckman-Singer estimates for alternative assumptions on
the individual-speciÞc effect.
Recall that the individual-speciÞc effect in this application is a proxy for the Þrm’s
optimal pricing rules. These pricing rules should be inherently correlated with the state
variables of the Þrm’s optimization problem, such as the rate of inßation and an indicator of industry demand. Serial correlation in these state variables would likely lead to
serial correlation in the pricing rules. Cecchetti assumes that the serial correlation is very
large and the innovations are very small, so that the Þrm’s optimal pricing rules can be
approximated by rules that are Þxed for non-overlapping 3-year periods.
In the data generating model used for the Monte Carlo exercises, the individual-speciÞc
effect was assumed to be drawn randomly from a uniform distribution. This assumption
conforms closely with the likelihood function of the Heckman-Singer model, which uses a
mass point distribution to represent random effects. If the individual-speciÞc effects are
generated under an alterative assumption, it is unclear how the Heckman-Singer model will
perform.
To investigate the properties of the Heckman-Singer model, additional Monte Carlo
exercises were undertaken. Instead of using a uniform distribution, the individual-speciÞc
effects are assumed to follow an autoregressive process of the form
ai,g = µ + ρai,g−1 + εi,g
(8)
where ai,g is the effect for Þrm i over the 3-year interval g and εi,g is a Gaussian innovation
to the process with mean 0 and standard deviation σε . In other words, there will be 9
21
individual-speciÞc 3-year effects created from the autoregressive process for the sample of
27 periods for each Þrm. The effects are assumed to be independent across Þrms.
Table A1 presents the results of Monte Carlo exercises when the individual-speciÞc
effects within a Þrm are generated by this autoregressive process. The exercise is repeated
for Þve different values of the persistence parameter, ρ ∈ {0, 0.3, 0.6, 0.9, 0.99}. The other
parameters, {µ, σε }, are speciÞed such that the mean of the process is -5 and the standard
deviation is 1. As expected, the Chamberlain conditional logit estimates fair poorly in all
cases due to the presence of state-dependent covariates. The logit estimates in column 3 are
downward biased due to the omitted variable problem, and the bias becomes slightly worse
for the duration and cumulative inßation coefficients as the persistence of the individualspeciÞc effect increases. The estimates from the Heckman-Singer model with 3 mass points
are closest to the true parameters. All three coefficient estimates decrease as persistence
increases, but the change in minimal. Interestingly, as the persistence increases, the fraction
of simulations in which the introduction of a fourth mass point would provide any additional
improvement in the likelihood decreases. Despite the misspeciÞcation of the likelihood
function in this case, the Heckman-Singer model still performs well.
Table A2 provides results of Monte Carlo exercises where there is no time dependence
in the data generating model. The results are qualitatively similar to Table A1, with the
exception that the Chamberlain conditional logit estimates are much more volatile.
One Þnal robustness exercise would be to examine the Heckman-Singer estimates when
the individual-speciÞc effects are correlated with aggregate inßation and industry demand
variables in the data generating model. In such a circumstance, it would be much more
effective to solve directly for the Þrm’s policy functions with a structural model than to
use an assumption of policy rules that are Þxed for 3-year intervals.
22
Table 1: Monte Carlo Exercise Based Upon Cecchetti’s Parameter Estimates
(1)
(2)
Truth Chamberlain
(3)
(4)
(5)
Logit
Heckman-Singer
Heckman-Singer
(2 mass points)
(3 mass points)
Cond. Logit
Ti,t,t̃
πi,t,t̃
Yú i,t,t̃
1.02
19.20
7.60
2.86
0.49
0.89
0.98
(1.20)
(0.06)
(0.15)
(0.19)
19.93
11.51
17.31
18.61
(13.19)
(1.51)
(2.85)
(3.36)
8.34
4.90
6.88
7.35
(5.00)
(1.12)
(1.67)
(1.88)
NOTE: The numbers reported in parentheses are standard deviations of the estimates across the 10,000
simulations. Estimates of the constant for the logit model and the mass points for the Heckman-Singer
speciÞcation are not displayed.
Table 2: Monte Carlo Exercise with No Time Dependence
(1)
(2)
Truth Chamberlain
(3)
(4)
Logit
Cond. Logit
Ti,t,t̃
πi,t,t̃
Yú i,t,t̃
0
19.20
7.60
(5)
Heckman-Singer Heckman-Singer
(2 mass points)
(3 mass points)
2.31
-0.02
0.00
0.00
(5.07)
(0.03)
(0.05)
(0.05)
29.83
11.26
18.00
19.92
(39.37)
(1.28)
(2.83)
(3.89)
6.26
5.10
7.25
7.84
(11.10)
(1.13)
(1.71)
(1.99)
NOTE: The numbers reported in parentheses are standard deviations of the estimates across the 10,000
simulations. Estimates of the constant for the logit model and the mass points for the Heckman-Singer
speciÞcation are not displayed.
23
Table 3: Coefficient Estimates for Magazine Price Model
(1)
(2)
(3)
Fixed-effects Chamberlain
Logit
Ti,t,t̃
πi,t,t̃
Yú i,t,t̃
Constant
(4)
Heckman-
Logit
Cond. Logit
Singer
i = mag.
i = 3-year
i = 3-year
-0.10
-0.07
1.02
-0.09
(0.03)
(0.03)
(0.28)
(0.04)
6.93
8.83
19.20
8.23
(1.12)
(1.25)
(7.51)
(1.53)
-0.36
-1.14
7.60
-0.13
(0.98)
(1.06)
(3.46)
(1.14)
-1.90
(0.14)
Mass point 1
Mass point 2
-1.94
[0.88]
(0.20)
[(0.16)]
-29.15
[0.12]
(1.1e+11) [(0.16)]
Log like.
-500.45
-473.18
-83.72
-499.65
Obs.
1026
1026
1026
1026
NOTE: The numbers in parentheses are standard errors, but estimates using Chamberlain’s method are
inconsistent as described above. The formulation used for specifying “individuals” in Þxed-effects estimations (choosing i) is stated at the top of each column. Column 1 reports unconditional logit estimates.
Column 2 reports unconditional logit estimates with Þxed-effects dummy variables. Coefficient estimates
for the dummy variables are not reported. Column 3 reports results from the Chamberlain conditional
logit estimation. Column 4 contains results from Heckman-Singer procedure where two mass points are
estimated to capture the individual-speciÞc effects. The probabilities associated with each mass point are
listed in square brackets.
24
Table 4: Coefficient Estimates when Jointly Controlling for Magazine and Year Effects
(1)
(2)
Logit
Logit with
Fixed Effects
Ti,t,t̃
πi,t,t̃
Yú i,t,t̃
Constant
-0.10
0.07
(0.03)
(0.07)
6.93
5.84
(1.12)
(2.25)
-0.36
-1.51
(0.98)
(2.23)
-1.90
(0.14)
Log like.
-500.45
-421.61
Obs.
1026
1026
NOTE: The numbers in parentheses are standard errors. Estimates presented in column 2 are from a
Þxed-effects unconditional logit speciÞcation which included yearly dummy variables and magazine dummy
variables. The coefficient estimates for the dummy variables are jointly signiÞcant and are omitted from
the table due to space considerations.
25
Table A1: Alternative Monte Carlo Exercises Based Upon Cecchetti’s Parameter Estimates
(1)
Truth
Ti,t,t̃
1.02
πi,t,t̃
19.20
Yú i,t,t̃
7.60
Ti,t,t̃
1.02
πi,t,t̃
19.20
Yú i,t,t̃
7.60
Ti,t,t̃
1.02
πi,t,t̃
19.20
Yú i,t,t̃
7.60
Ti,t,t̃
1.02
πi,t,t̃
19.20
Yú i,t,t̃
7.60
Ti,t,t̃
1.02
πi,t,t̃
19.20
Yú i,t,t̃
7.60
(2)
Chamberlain
Cond. Logit
3.18
(1.65)
14.45
(13.29)
8.32
(5.47)
3.19
(1.41)
14.13
(12.99)
8.33
(5.36)
3.20
(1.60)
13.91
(12.87)
8.34
(5.36)
3.17
(1.08)
13.84
(12.68)
8.42
(5.30)
3.20
(1.75)
13.82
(13.06)
8.43
(5.45)
(3)
Logit
0.78
(0.07)
15.77
(1.72)
6.33
(1.20)
0.77
(0.08)
15.72
(1.78)
6.34
(1.21)
0.76
(0.08)
15.68
(1.82)
6.34
(1.19)
0.75
(0.08)
15.66
(1.74)
6.34
(1.13)
0.75
(0.09)
15.65
(1.66)
6.35
(1.10)
(4)
Heckman-Singer
(2 mass points)
0.86
(0.16)
16.96
(2.36)
6.61
(1.66)
0.84
(0.15)
16.84
(2.37)
6.59
(1.67)
0.82
(0.15)
16.69
(2.33)
6.54
(1.62)
0.82
(0.14)
16.65
(2.25)
6.55
(1.54)
0.82
(0.14)
16.65
(2.17)
6.56
(1.52)
(5)
Heckman-Singer
(3 mass points)
0.99
(0.18)
18.85
(2.91)
7.41
(1.70)
0.97
(0.18)
18.64
(2.94)
7.37
(1.70)
0.95
(0.17)
18.41
(2.89)
7.31
(1.63)
0.93
(0.17)
18.22
(2.74)
7.26
(1.56)
0.93
(0.17)
18.23
(2.70)
7.27
(1.55)
ρ=0
ρ = 0.3
ρ = 0.6
ρ = 0.9
ρ = 0.99
NOTE: The numbers reported in parentheses are standard deviations of the estimates across the 10,000
simulations.
26
Table A2: Alternative Monte Carlo Exercises with No Time Dependence
(1)
Truth
Ti,t,t̃
0
πi,t,t̃
19.20
Yú i,t,t̃
7.60
Ti,t,t̃
0
πi,t,t̃
19.20
Yú i,t,t̃
7.60
Ti,t,t̃
0
πi,t,t̃
19.20
Yú i,t,t̃
7.60
Ti,t,t̃
0
πi,t,t̃
19.20
Yú i,t,t̃
7.60
Ti,t,t̃
0
πi,t,t̃
19.20
Yú i,t,t̃
7.60
(2)
Chamberlain
Cond. Logit
6.05
(16.66)
31.03
(160.02)
8.72
(88.15)
6.10
(16.89)
25.93
(143.93)
9.52
(89.82)
6.40
(18.92)
24.97
(143.52)
15.81
(104.68)
7.10
(21.58)
17.05
(107.99)
14.38
(98.23)
7.64
(25.01)
16.67
(156.08)
15.40
(97.87)
(3)
Logit
-0.01
(0.03)
15.70
(1.50)
6.50
(1.25)
-0.02
(0.03)
15.50
(1.53)
6.38
(1.24)
-0.03
(0.03)
15.31
(1.54)
6.28
(1.17)
-0.04
(0.03)
15.12
(1.56)
6.17
(1.20)
-0.04
(0.03)
15.06
(1.59)
6.08
(1.15)
(4)
Heckman-Singer
(2 mass points)
0.00
(0.04)
18.59
(3.14)
7.34
(1.58)
-0.01
(0.04)
18.12
(3.10)
7.12
(1.54)
-0.02
(0.03)
17.61
(2.99)
6.95
(1.46)
-0.03
(0.03)
17.20
(3.07)
6.75
(1.45)
-0.04
(0.03)
16.98
(2.97)
6.58
(1.42)
(5)
Heckman-Singer
(3 mass points)
0.00
(0.04)
20.47
(3.90)
7.99
(1.83)
-0.01
(0.04)
19.78
(3.79)
7.71
(1.80)
-0.02
(0.04)
19.10
(3.98)
7.45
(1.73)
-0.03
(0.04)
18.46
(3.86)
7.19
(1.67)
-0.04
(0.03)
18.19
(3.59)
7.02
(1.63)
ρ=0
ρ = 0.3
ρ = 0.6
ρ = 0.9
ρ = 0.99
NOTE: The numbers reported in parentheses are standard deviations of the estimates across the 10,000
simulations.
27